The existence and the continuity of the transition density function of the diffusion process generated by the Baouendi–Grushin operator is shown with the help of the partial Malliavin calculus. For this purpose, the partial Malliavin calculus is reformulated in terms of Watanabe’s distribution theory on Wiener spaces.
{"title":"AN APPLICATION OF THE PARTIAL MALLIAVIN CALCULUS TO BAOUENDI-GRUSHIN OPERATORS","authors":"S. Taniguchi","doi":"10.2206/kyushujm.73.417","DOIUrl":"https://doi.org/10.2206/kyushujm.73.417","url":null,"abstract":"The existence and the continuity of the transition density function of the diffusion process generated by the Baouendi–Grushin operator is shown with the help of the partial Malliavin calculus. For this purpose, the partial Malliavin calculus is reformulated in terms of Watanabe’s distribution theory on Wiener spaces.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68556493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. The manifold M of star-shaped curves in R n is considered via the theory of connections on vector bundles, and cyclic D -modules. The appropriate notion of an ‘integral curve’ (i.e. certain admissible deformations) on M is defined, and the resulting space of admissible deformations is classified via iso-spectral flows, which are shown to be described by equations from the n -KdV (Korteweg–de Vries) hierarchy.
{"title":"ON THE GEOMETRY OF STAR-SHAPED CURVES IN Rn","authors":"Stefan A. HOROCHOLYN","doi":"10.2206/kyushujm.73.123","DOIUrl":"https://doi.org/10.2206/kyushujm.73.123","url":null,"abstract":". The manifold M of star-shaped curves in R n is considered via the theory of connections on vector bundles, and cyclic D -modules. The appropriate notion of an ‘integral curve’ (i.e. certain admissible deformations) on M is defined, and the resulting space of admissible deformations is classified via iso-spectral flows, which are shown to be described by equations from the n -KdV (Korteweg–de Vries) hierarchy.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68555590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be 3. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar criteria for the class number of such fields to be 2 and 3.
{"title":"A NOTE ON CERTAIN REAL QUADRATIC FIELDS WITH CLASS NUMBER UP TO THREE","authors":"K. Chakraborty, Azizul Hoque, Mohit Mishra","doi":"10.2206/kyushujm.74.201","DOIUrl":"https://doi.org/10.2206/kyushujm.74.201","url":null,"abstract":"We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be 3. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar criteria for the class number of such fields to be 2 and 3.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68557353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present explicit formulas for all finite multiple zeta values by introducing a multiple generalization of Bernoulli polynomials associated with finite multiple zeta values. Furthermore we show that these values are also described by special values of multiple zeta functions and multiple star analogues of the Hurwitz zeta function.
{"title":"FINITE MULTIPLE ZETA VALUES, MULTIPLE ZETA FUNCTIONS AND MULTIPLE BERNOULLI POLYNOMIALS","authors":"Y. Komori, 小森 靖, ヤスシ コモリ","doi":"10.2206/KYUSHUJM.72.333","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.72.333","url":null,"abstract":"We present explicit formulas for all finite multiple zeta values by introducing a multiple generalization of Bernoulli polynomials associated with finite multiple zeta values. Furthermore we show that these values are also described by special values of multiple zeta functions and multiple star analogues of the Hurwitz zeta function.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"72 1","pages":"333-342"},"PeriodicalIF":0.4,"publicationDate":"2018-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46503494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positive Ricci curvature.
{"title":"A CONSTRUCTION OF GRAPHS WITH POSITIVE RICCI CURVATURE","authors":"Taiki Yamada","doi":"10.2206/kyushujm.74.291","DOIUrl":"https://doi.org/10.2206/kyushujm.74.291","url":null,"abstract":"Two complete graphs are connected by adding some edges. The obtained graph is called the gluing graph. The more we add edges, the larger the Ricci curvature on it becomes. We calculate the Ricci curvature of each edge on the gluing graph and obtain the least number of edges that result in the gluing graph having positive Ricci curvature.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45651196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove an identity for multiple zeta star values, which generalises some identities due to Imatomi, Tanaka, Tasaka and Wakabayashi. This identity gives an analogue of cyclic insertion type identities, for multiple zeta star values, and connects the block decomposition with Zhao's generalised 2-1 formula.
{"title":"ANALOGUES OF CYCLIC INSERTION-TYPE IDENTITIES FOR MULTIPLE ZETA STAR VALUES","authors":"Steven Charlton","doi":"10.2206/kyushujm.74.337","DOIUrl":"https://doi.org/10.2206/kyushujm.74.337","url":null,"abstract":"We prove an identity for multiple zeta star values, which generalises some identities due to Imatomi, Tanaka, Tasaka and Wakabayashi. This identity gives an analogue of cyclic insertion type identities, for multiple zeta star values, and connects the block decomposition with Zhao's generalised 2-1 formula.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42965387","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In part one we prove a theorem about the automorphism of solutions to Ramanujan's differential equations. We also investigate possible applications of the result. In part two we prove a similar theorem about the automorphism of solutions to the first-order system of differential equations associated to the generalised Chazy equation with parameter $k=frac{3}{2}$.
{"title":"AUTOMORPHISM OF SOLUTIONS TO RAMANUJAN'S DIFFERENTIAL EQUATIONS AND OTHER RESULTS","authors":"Matthew Randall","doi":"10.2206/KYUSHUJM.75.77","DOIUrl":"https://doi.org/10.2206/KYUSHUJM.75.77","url":null,"abstract":"In part one we prove a theorem about the automorphism of solutions to Ramanujan's differential equations. We also investigate possible applications of the result. In part two we prove a similar theorem about the automorphism of solutions to the first-order system of differential equations associated to the generalised Chazy equation with parameter $k=frac{3}{2}$.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44549897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Two explicit sets of solutions to the double shuffle equations modulo products were introduced by Ecalle and Brown respectively. We place the two solutions into the same algebraic framework and compare them. We find that they agree up to and including depth four but differ in depth five by an explicit solution to the linearized double shuffle equations with an exotic pole structure.
{"title":"ON ECALLE'S AND BROWN'S POLAR SOLUTIONS TO THE DOUBLE SHUFFLE EQUATIONS MODULO PRODUCTS","authors":"Nils Matthes, K. Tasaka","doi":"10.2206/kyushujm.73.337","DOIUrl":"https://doi.org/10.2206/kyushujm.73.337","url":null,"abstract":"Two explicit sets of solutions to the double shuffle equations modulo products were introduced by Ecalle and Brown respectively. We place the two solutions into the same algebraic framework and compare them. We find that they agree up to and including depth four but differ in depth five by an explicit solution to the linearized double shuffle equations with an exotic pole structure.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46875653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isomorphism between Jacobi forms of integral weight and half-integral weight is also obtained.
{"title":"RANKIN-SELBERG METHOD FOR JACOBI FORMS OF INTEGRAL WEIGHT AND OF HALF-INTEGRAL WEIGHT ON SYMPLECTIC GROUPS","authors":"S. Hayashida","doi":"10.2206/kyushujm.73.391","DOIUrl":"https://doi.org/10.2206/kyushujm.73.391","url":null,"abstract":"In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of Fourier-Jacobi coefficients of two Jacobi cusp forms. The denominators and the range of summation of these Dirichlet series are like the ones of the Koecher-Maass series. The meromorphic continuations and functional equations of these Dirichlet series are obtained. Moreover, an identity between the Petersson norms of Jacobi forms with respect to linear isomorphism between Jacobi forms of integral weight and half-integral weight is also obtained.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2018-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46370742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper, asymptotic behavior of the uncertainty product for a family of zonal spherical wavelets is computed. The family contains the most popular wavelets, such as Gauss-Weierstrass, Abel-Poisson and Poisson wavelets and Mexican needlets. Boundedness of the uncertainty constant is in general not given, but it is a property of some of the wavelets from this class.
{"title":"ON THE UNCERTAINTY PRODUCT OF SPHERICAL WAVELETS","authors":"I. Iglewska-Nowak","doi":"10.2206/kyushujm.71.407","DOIUrl":"https://doi.org/10.2206/kyushujm.71.407","url":null,"abstract":"In the paper, asymptotic behavior of the uncertainty product for a family of zonal spherical wavelets is computed. The family contains the most popular wavelets, such as Gauss-Weierstrass, Abel-Poisson and Poisson wavelets and Mexican needlets. Boundedness of the uncertainty constant is in general not given, but it is a property of some of the wavelets from this class.","PeriodicalId":49929,"journal":{"name":"Kyushu Journal of Mathematics","volume":"71 1","pages":"407-416"},"PeriodicalIF":0.4,"publicationDate":"2018-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43872990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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